Difficult Danny 2

Difficult Danny has acquired his nickname (and some other, less polite nicknames) due to the fact that, whenever he buys anything, he insists upon paying the exact amount, with precisely 100 pieces of currency. Danny wanted an xCube365 (it lasts 365 days before programmatically disabling itself, so that a new one must be purchased), the highly coveted gaming system that had just been released. It cost $230, but he managed to save up enough to buy one. He paid using some quantity (possibly zero) of each the following denominations of bills: $1, $5, $10, $20, $50, and $100. Of those six quantities (including any zeros), there is one -- and ONLY one -- distinct triplet that can be made into a true multiplication sentence. For example, if there were 2 ones, 3 twenties, and 6 tens, 2 x 3 = 6, so this would fit the rule. How many of each denomination did Danny use to buy the xCube365?

Answer

Can at least one of each denomination be used? That totals $186 with only 6 bills, plus at least $94 with the rest of the bills. This is too high. Can two or more of the denominations not be used? Any number multiplied by 0 is 0. Therefore, if there are two or more 0 quantities, any of the other three could be multiplied by one of the 0's, using the other 0 for the result. This means that there would be more than one triplet that fits the rule. So we know that exactly 5 of the 6 denominations are used.

If a $100 bill is used, there are only two combinations that have the proper total quantity of bills and dollar sum, but they both involve two 0-quantities. So $100 bill is the denomination that is not used. We now know that at least one of each of the other bills is used. That totals $86 with 5 bills, leaving 95 bills to make the other $144.

There are only 4 ways to make $144 with these denominations, which are listed in order from $1 bills to $50 bills in each of these sets: {84,10,1,0,0}, {89,1,5,0,0}, {89,3,2,1,0}, {94,0,0,0,1}. Adding in one of each bill, which we have already determined, gives {85,11,2,1,1}, {90,2,6,1,1}, {90,4,3,2,1}, {95,1,1,1,2}. {1,1,1} is the only triplet here that can make a true multiplication sentence: 1 x 1 = 1. Therefore, Danny paid with 95 $1 bills, a $5 bill, a $10 bill, a $20 bill, and 2 $50 bills.Hide